Question on Parabola

Topic : Parabola

Question : Natural bridge is a stone arch formed over a river or stream. The longest natural bridge in the world is rainbow bridge in Utah. if the origin is placed at one end of the arch, the cure of the arch can be modeled by the equation h=-0.042d^2+3.57d where h meters represents the height and d meters represents the horizontal distance.
a. what is the max height of the arch, to the nearest meter?
b. at a horizontal distance of 10 m from the vertex, what is the height of the arch, to the nearest meter

Answer :

H(d) = -0.042d^2 + 3.57d
where h(d) represents the height of the arch and d represents the horizontal distance from one end of the base of the arch in meters. Find the maximum height of the arch.
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Max and min problems use the 1st derivative of the function..
h(d) = -0.042d^2 + 3.57d
Set the 1st derivative to zero.
0 = -0.084d + 3.57
0.084d = 3d = 42.5 meters
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h(42.5) = -0.042*(42.5^2 )+ 3.57*(42.5)= -75 .8625+ 151.725h = 75.8625 meters = 76 appx..
At a horizontal distance of 10 m from the vertex, what is the height of the arch, to the nearest meter
h(d) = -0.042d^2 + 3.57d
h(10) = -0.042*10^2 + 3.57*10
=-4.2+35.7=31.5 meters